quote:Original post by FReY
Yes, you using the term ''vector'' as the semantic definition of offset-vector from an origin.

The terminology I am using is the standard for Affine Space.
http://mathworld.wolfram.com/AffineSpace.html

quote:
If you really think about it, points are offset-vectors but just from the origin (our 3d space reference point).

I guess what you mean is that once you select a point as origin (which is an arbitrary selection), you can represent a point by the vector that you have to add to the origin to get it. True, but doesn''t change anything I said.

quote:
It is easier to conceptualize your rules written as follows:

- add 2 direction vectors, getting a direction vector
- multiply a direction vector by a scale, getting a scaled direction vector
- add a position-vector and a direction-vector, getting a position-vector
etc...

You can come up with a new language if you want, but "point" and "vector" are the standard ways of saying it. On top of that, I would not like calling points "position-vectors", because the word "vector" implies that those things can be added and multiplied by scalars, which is not true of points.

quote:
But I prefer to think of vectors as just arrays, not without any semantic meaning behind them.

I can see that. In practical terms, there are good arguments for and against having separate classes for points and vectors.
FOR:
- It more closely reflects the mathematical structures that we are using.
- It prevents some mistakes in compile time (adding two points will give an error, as that operation is undefined).
AGAINST:
- Expressions like "midpoint = .5*(point_A + point_B);" are not allowed, although the result is perfectly well defined. You would have to re-write that as "midpoint = point_A + .5*(point_B - point_A);", which is probably a little slower to evaluate.

Of course points (position vectors) can be multiplied/divided by scalars or other vectors... Adding a position vector to another is just as valid and common. What else are you doing when you''re translating or rotating or scaling a 3d object? A transformation matrix is merely a convenient way to package together the vectors and scalars you''re adding and multiplying with.

Related to the original post, "distance" or difference between two (classical) vectors can also be a very useful concept. For example, in orbital mechanics you will often see people talking about "delta-v" which is the difference between two velocity vectors usually measured in m/s and calculated as |v2-v1| .

quote:Original post by Fingers_
Of course points (position vectors) can be multiplied/divided by scalars or other vectors... Adding a position vector to another is just as valid and common. What else are you doing when you''re translating or rotating or scaling a 3d object? A transformation matrix is merely a convenient way to package together the vectors and scalars you''re adding and multiplying with.

When you translate or rotate or scale a 3d object you are applying an affine transformation, which a well defined operation. What I mean by "well defined" is that if your transformation does not depend on your choice of coordinates.

In a translation, you are adding a point and a vector, and you get a point. That was one of the allowed operations.

Multiplying by a scalar is not a well defined operation. If you change the origin, you get a different transformation.

quote:Original post by Fruny Quite a few bases exist. That''s the theoretical fundation for Fourier transforms, Laplace transforms, wavelet transforms, Z transforms and many more (Mellin, Hankel, Gabor...)

Those are not really bases of the vector spaces. It is not true that any function can be expressed as a linear combination of a finite number of members of the base.

All your proposed "bases" are numerable. A base of the set of continuous real functions, for example, cannot be numerable. The functions F_a(x) = |x - a|, where a is a real number, form a linearly independent set. Any linearly independent set can be expanded to form a base, so there is a base that is not numerable. And all bases have the same cardinal.

Based on the OPs very simple question, this discussion is waaaaaay off topic. As people have stated, yes... a vector space is a member of any member of a vector space see here for more details. You cannot really call points postitional vectors, as a many "normal" spaces would not necessarily fulfill the criterion of being a vector space. It is true however that most "normal" spaces are subsets of some vector space. As for bases, any vector space can have a basis, whether this basis is finite of not, is another matter entirely. I suspect the OP just wanted confirmation that their formula for distance between two Verticies formula was correct... which is was, so to the OP, You are correct .

PS: A vector space can be made over any field, and hence they are VERY general constructs. All you need is to fulfil the vector space criterion, and away we go. I think we should all assume that when someone says vector, unless otherwise specified... they mean a (1,2,3 or 4 dimensional usually) vector over the reals. If they want to refer to STL vectors (which are abstract vectors, over the template classes field) then they should qualify it. Just a suggestion, to try and cut down on ambiguity in future.